PowerPoint created by Parsheena Berch Resource : JBHM material Pictures: Google Images

Equations and Inequalities

How well do you balance with math? Explain why the scale is not balanced How do I make the scale balance? Today’s lesson will focus on equations and inequalities. It is important for you to remember that the key is balance. The two sides must always balance or be equal just like the water on the balance scale.

Page 128 __ + 2 = 5 What number goes in the blank? x + 2 = 5 What is different about these two sentences? The only difference is one uses a blank and one uses an x to take the place of the number 3. Variable – a symbol such as a letter, box, star, question mark, etc. that is used to represent an unknown or undetermined value in an expression or equation.

Can you find the variables? y = 573 z – 24 = 13 a ÷ 42 = 6 ? 4 = 48 Variables can take different forms, letters, symbols, etc. but they are still variables and the process to solve or determine values is the same.

Equation – a statement that two mathematical expressions are equal x + 3 = 12 y – 2.8 = = 7 + z 2 ½ + y = 6 Write an equation on your own.

Before going any farther and beginning to work with equations, you need to learn some different methods of writing some of the operations you have been working with for years. You are going to learn some new ways to show multiplication and division without using the traditional operational symbols x and ÷.

Multiplication: 2 x 5 = = 10 2x = 10 2(5) = 10 (You should not use the traditional symbol, x, for a multiplication sign when working with equations and variables as it can easily be mistaken for a variable rather than multiplication sign.) Division: 10 ÷ 2 = 5 10/5 = 2

Write the equation that I say on your white board. fourteen divided by a equals two (14 / a = 2 or use the division symbol) X divided by twenty equals five (x ÷ 20 = 5 x/20 = 5) y times six equals eighteen (6y = 18 6 y = 18) Four times y equals twelve (4y = 12 4 y = 12)

Solving Equations: When working with equations, you will be working to solve for the variable. This means you will be working to determine the value of the variable or what number the variable is taking the place of. There are different methods that can be used to solve equations and you will be learning multiple strategies. Then you will be able to decide which strategy or method you are most comfortable working with.

Inverse Operations (Page 128 at the bottom) Addition and Subtraction Tamatha was planning her birthday party, and she wanted to use lots of balloons. Because her cake was going to be purple and blue, she decided on these two colors for her balloons. If she needed 125 balloons and she bought 45 purple balloons, how many blue balloons would she need to buy? (Page 129 in student binder of JBHM) Brainstorm and give me your ideas. She would need 80 blue balloons because 125 total balloons minus 45 purple balloons leave 80 blue balloons. Go a step further and see if students can match this equation to the story. Example: b + 45 = 125 This equation says that 45 (purple balloons) plus (b) some unknown number (blue balloons) = 125 total balloons.

b + 45 = 125 If you had been given this equation first without the story, how would you have solved it? Look at your Binder Notes and the chart that shows the addition operation with the inverse being subtraction. Inverse means opposite. The equal sign means that the sides are to always remain the same so the equation balances. An equation is a statement that two mathematical expressions are equal. I need two students who are the same height. Now let us go back to the balloon story. b + 45 = 125 The answer reveals the number of blue balloons needed.

Example of Balloon Problem Solution: b + 45 = 125 (addition operation) b + 45 – 45 = 125 – 45 (inverse operation of subtraction) b = 80 (how many blue balloons are needed)

Substitute the answer in the equation and check for accuracy. b + 45 = = = 125

This is an inverse operation. Inverse operations are opposite operations. Addition is the opposite of subtraction. All equations are solved using inverse operations. An inverse operation “undoes” an operation and leaves the unknown weight (variable) standing alone or isolated.

Example #1 x + 3 = 12 Operation – addition x + 3 – 3 = 12 – 3 Inverse operation – subtraction x = 9 Check your work! x + 3 = 12 (9) + 3 = 12

Example #2 x – 3 = 12 Operation – subtraction x – = Inverse operation – addition x = 15 Check your work! x – 3 = 12 (15) – 3 = 12

Example #3 x – 3.5 = 10 Operation – subtraction x – = Inverse operation - addition x = 13.5 Check your work! x – 3.5 = 10 (13.5) – 3.5 = 10

Example #4 (Page 130) x = 12 Operation – addition x – 4.6 = 12 – 4.6 Inverse operation – subtraction x = 7.4 Check your work! x = 12 (7.4) = 12

Example #5 x – ½ = 4 Operation – subtraction X – ½ + ½ = 4 + ½ Inverse operation – addition X = 4 ½ Check your work! X – ½ = 4 (4 ½) – ½ = 4

Example #6 x + 3 ¼ = 5 3/4 operation – addition X + 3 ¼ - 3 ¼ = 5 ¾ - 3 ¼ Inverse operation – subtraction X = 2 ½ Check your work! x + 3 ¼ = 5 ¾ (2 1/2) + 3 ¼ = 5 ¾

Work these on page 131 x + 8 = 19 x – 4 = 42 x – 22 = 12 x = 10 x – 4.9 = 11 x + ¾ = 8

Closure: Explain what you have learned about equations today. Does the (Left side = right side) always?

Homework: 1 st, 2 nd, and 7 th You HAVE to check your work! x + 25 = y = 89 a – 14 = 75 x – 9 = -7

Homework: 3 rd, 4 th, and 6 th You HAVE to check your work! x + 5 = 16 x – 4 = x = x = 20